Planetary mass

The four terrestrial planets in the Solar System, to scale. Note that Mars (to the far right) is about half the diameter of the Earth, but only one-ninth of the mass.

The four gas giants in the Solar System, shown to scale against a limb of the Sun. The Sun is by far the largest body in the Solar System, which makes it convenient to use the solar mass as a reference for planetary masses. The terrestrial planets would not be visible at this scale.

Planetary mass is a measure of the mass of a planet. Within the Solar System, planets are usually measured in the astronomical system of units, where the unit of mass is the solar mass, the mass of the Sun. In the study of extrasolar planets, the unit of measure is typically the mass of Jupiter for large gas giant planets, and the mass of Earth for smaller rocky terrestrial planets.

The mass of a planet within the Solar System is an adjusted parameter in the preparation of ephemerides. There are two basic procedures for measurement:

• If the planet has natural satellites, its mass can be calculated from Newton's law of universal gravitation (with use of Kepler's third law to simplify calculations changing an elliptical orbit by circle one). The same procedure can be used for data from space probes such as the Voyager probes to the outer planets and the MESSENGER spacecraft to Mercury.
• The mass of a planet can also be inferred from its effect on the orbits of other planets.

In practice, both methods are used. The ephemeris is a model of the Solar System, and the planetary masses within that model are adjusted so as to give the best fit between the model and the observed positions of the planets.

Choice of units

The choice of the solar mass, M_☉, as the basic unit for planetary mass comes directly from the calculations used to determine planetary mass. In the most precise case, that of the Earth itself, the mass is known in terms of solar masses to twelve significant figures: the same mass, in terms of kilograms or other Earth-based units, is only known to five significant figures, that is more than a million times less precise.^[1]

The difference comes from the way in which planetary masses are calculated. It is impossible to "weigh" a planet, and much less the Sun, against the sort of mass standards which are used in the laboratory. On the other hand, the orbits of the planets give a great range of observational data as to the relative positions of each body, and these positions can be compared to their relative masses using Newton's law of universal gravitation (with small corrections for General Relativity where necessary). To convert these relative masses to Earth-based units such as the kilogram, it is necessary to know the value of the Newtonian gravitational constant, G. This constant is remarkably difficult to measure in practice, and its value is only known to a precision of one part in ten-thousand.^[2]

The solar mass is quite a large unit on the scale of the solar system: 1.9884(2)×10³⁰ kg.^[1] The largest planet, Jupiter, is 0.09% the mass of the Sun, while the Earth is about three millionths (0.0003%) of the mass of the Sun. Various different conventions are used in the literature to overcome this problem: for example, inverting the ratio so that one quotes the planetary mass in the 'number of planets' it would take to make up one Sun.^[1] Here, we have chosen to list all planetary masses in 'microSuns' – that is the mass of the Earth is just over three 'microSuns', or three millionths of the mass of the Sun – unless they are specifically quoted in kilograms.

	Mass relative to
	the Earth	Jupiter
Mercury	0.0553	0.000174
Venus	0.815	0.00256
Earth	1	0.00315
Mars	0.107	0.000338
Jupiter	318	1
Saturn	95.2	0.299
Uranus	14.5	0.0457
Neptune	17.2	0.0540

When comparing the planets among themselves, it is often convenient to use the mass of the Earth (M_E or M_⊕) as a standard, particularly for the terrestrial planets. For the mass of gas giants, and also for most extrasolar planets and brown dwarfs, the mass of Jupiter (M_J) is a convenient comparison.

Vesta is the second largest body in the asteroid belt after Ceres. This image from the Hubble Space Telescope shows that it is far from being spherical.

Planetary mass and planet formation

The mass of a planet has consequences for its structure, especially while it is in the process of formation. A body which is more than about one ten-thousandth of the mass of the Earth can overcome its compressive strength and achieve hydrostatic equilibrium: it will be roughly spherical, and since 2006 has been classified as a dwarf planet if it orbits around the Sun (that is, if it is not the satellite of another planet). Smaller bodies like asteroids are classified as "small Solar System bodies".

A dwarf planet, by definition, is not massive enough to have gravitationally cleared its neighbouring region of planetesimals: it is not known quite how large a planet must be before it can effectively clear its neighbourhood, but one tenth of the Earth's mass is certainly sufficient.

The smaller planets retain only silicates, and are terrestrial planets like Earth or Mars, though multiple-M_E super-Earths have been discovered. The interior structure of rocky planets is mass-dependent: for example, plate tectonics may require a minimum mass to generate sufficient temperatures and pressures for it to occur.^[3]

If the protoplanet grows by accretion to more than about 5–10 times the mass of the Earth, its gravity become large enough to retain hydrogen in its atmosphere. In this case, it will grow into a gas giant. If the planet then begins migration, it may move well within its system's snow line, and become a hot Jupiter orbiting very close to its star, then gradually losing small amounts of mass as the star's radiation strips its atmosphere.

The theoretical minimum mass a star can have, and still undergo hydrogen fusion at the core, is estimated to be about 75 times the mass of Jupiter, though fusion of deuterium can occur at masses as low as 13 Jupiters.^[4][5][6]

Values from the DE405 ephemeris

The DE405/LE405 ephemeris from the Jet Propulsion Laboratory^[1][7] is a widely used ephemeris dating from 1998 and covering the whole Solar System. As such, the planetary masses form a self-consistent set, which is not always the case for more recent data (see below).

		Planetary mass × 10-6 (relative to the Sun)	Satellite mass (relative to the parent planet)	Absolute mass	Mean density
Planets and natural satellites
Mercury		0.16601		3.301×1023 kg	5.43 g/cm3
Venus		2.4478383		4.867×1024 kg	5.24 g/cm3
Earth/Moon system		3.04043263333		6.046×1024 kg	4.4309
	Earth	3.00348959632		5.972×1024 kg
	Moon		1.23000383×10−2	7.346×1022 kg
Mars		0.3227151		6.417×1023 kg	3.91 g/cm3
Jupiter		954.79194		1.899×1027 kg	1.24 g/cm3
	Io		4.70×10−5	8.93×1022 kg
	Europa		2.53×10−5	4.80×1022 kg
	Ganymede		7.80×10−5	1.48×1023 kg
	Callisto		5.67×10−5	1.08×1023 kg
Saturn		285.8860		5.685×1026 kg	0.62 g/cm3
	Titan		2.37×10−4	1.35×1023 kg
Uranus		43.66244		8.682×1025 kg	1.24 g/cm3
	Titania		4.06×10−5	3.52×1021 kg
	Oberon		3.47×10−5	3.01×1021 kg
Neptune		51.51389		1.024×1026 kg	1.61 g/cm3
	Triton		2.09×10−4	2.14×1022 kg
Dwarf planets and asteroids
Pluto/Charon system		0.007396		1.471×1022 kg	2.06 g/cm3
Ceres		0.00047		9.3×1020 kg
Vesta		0.00013		2.6×1020 kg
Pallas		0.00010		2.0×1020 kg

Earth mass and lunar mass

Where a planet has natural satellites, its mass is usually quoted for the whole system (planet + satellites), as it is the mass of the whole system which acts as a perturbation on the orbits of other planets. The distinction is very slight, as natural satellites are much smaller than their parent planets (as can be seen in the table above, where only the largest satellites are even listed).

The Earth and the Moon form a case in point, partly because the Moon is unusually large (just over 1% of the mass of the Earth) in relation to its parent planet compared with other natural satellites. There are also very precise data available for the Earth–Moon system, particularly from the Lunar Laser Ranging Experiment (LLR).

The geocentric gravitational constant – the product of the mass of the earth times the Newtonian gravitational constant – can be measured to high precision from the orbits of the Moon and of artificial satellites. The ratio of the two masses can be determined from the slight wobble in the Earth's orbit caused by the gravitational attraction of the moon.

More recent values

The construction of a full, high-precision Solar System ephemeris is an onerous task.^[8] It is possible (and somewhat simpler) to construct partial ephemerides which only concern the planets (or dwarf planets, satellites, asteroids) of interest by "fixing" the motion of the other planets in the model. The two methods are not strictly equivalent, especially when it comes to assigning uncertainties to the results: however, the "best" estimates – at least in terms of quoted uncertainties in the result – for the masses of minor planets and asteroids usually come from partial ephemerides.

Nevertheless, new complete ephemerides continue to be prepared, most notably the EPM2004 ephemeris from the Institute of Applied Astronomy of the Russian Academy of Sciences. EPM2004 is based on 317014 separate observations between 1913 and 2003, more than seven times as many as DE405, and gave more precise masses for Ceres and five asteroids.^[8]

Planetary mass × 10^-6 (relative to the Sun)

	EPM2004[8]	Vitagliano & Stoss (2006)[9]	Brown & Schaller (2007)[10]	Tholen et al. (2008)[11]	Pitjeva & Standish (2009)[12]	Ragozzine & Brown (2009)[13]
136199 Eris			84.0(1.0)×10−4
134340 Pluto				73.224(15)×10−4 [note 1]
136108 Haumea						20.1(2)×10−4
1 Ceres	4.753(7)×10−4				4.72(3)×10−4
4 Vesta	1.344(1)×10−4				1.35(3)×10−4
2 Pallas	1.027(3)×10−4				1.03(3)×10−4
15 Eunomia		0.164(6)×10−4
3 Juno	0.151(3)×10−4
7 Iris	0.063(1)×10−4
324 Bamberga	0.055(1)×10−4

IAU current best estimates (2009)

A new set of "current best estimates" for various astronomical constants^[14] was approved the 27th General Assembly of the International Astronomical Union (IAU) in August 2009.^[15] It includes masses for all the planets except the Earth–Moon system, as well as Eris, Pluto, Ceres, Vesta and Pallas: values for the masses of Eris, Pluto, Ceres, Vesta and Pallas are as given in the table above. Except for those of Mercury and Uranus, all the planetary masses have been revised since the DE405 ephemeris (1998).

	Ratio of the solar mass to the planetary mass	Planetary mass × 10-6 (relative to the Sun)	Mass/kg	Ref
Mercury	6023.6(3)×103	0.166014(8)	3.3010(3)×1023	[16]
Venus	408.523719(8)×103	2.08106272(3)	4.1380(4)×1024	[17]
Mars	3098.70359(2)×103	0.3232371722(21)	6.4273(6)×1023	[18]
Jupiter [note 2]	1.0473486(17)×103	954.7919(15)	1.89852(19)×1027	[19]
Saturn	3.4979018(1)×103	285.885670(8)	5.6846(6)×1026	[20]
Uranus	22.90298(3)×103	43.66244(6)	8.6819(9)×1025	[21]
Neptune	19.41226(3)×103	51.51384(8)	1.02431(10)×1026	[22]

The ratio of the mass of the Moon to the mass of the Earth is given as 1.23000371(4)×10⁻²,^[12] while the ratio of the mass of the Sun to the mass of the Earth can be calculated as the ratio of the heliocentric and geocentric gravitational constants: 332.9460487(7)×10³, giving the mass of the Earth as 3.003486962(6)×10⁻⁶ M_☉ or 5.9722(6)×10²⁴ kg.^[23]

Notes

1. ^ For ease of comparison with other values, the mass given in the table is for the entire Pluto system: this is also the value which appears in the IAU "current best estimates". Tholen et al. also give estimates for the masses of the four bodies which comprise the Pluto system: Pluto 6.558(28)×10⁻⁹ M_☉, 1.304(5)×10²² kg; Charon 7.64(21)×10⁻¹⁰ M_☉, 1.52(4)×10²¹ kg; Nix 2.9×10⁻¹³ M_☉, 5.8×10¹⁷ kg; Hydra 1.6×10⁻¹³ M_☉, 3.2×10¹⁷ kg.
2. ^ The value quoted by the IAU Working Group on Numerical Standards for Fundamental Astronomy (1.047348644×10³) is inconsistent with the quoted uncertainty (1.7×10⁻³): the value has been rounded here.

References

1. ^ ^{a b c d} "2009 Selected Astronomical Constants" in The Astronomical Almanac Online, USNO–UKHO, http://asa.usno.navy.mil/ .
2. ^ Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80: 633–730. Bibcode 2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633. http://physics.nist.gov/cuu/Constants/codata.pdf.  Direct link to value..
3. ^ CfA Press Release Release No.: 2008-02 January 09, 2008 Earth: A Borderline Planet for Life?
4. ^ Boss, Alan (2001-04-03), Are They Planets or What?, Carnegie Institution of Washington, archived from the original on 2006-09-28, http://web.archive.org/web/20060928065124/http://www.carnegieinstitution.org/News4-3,2001.html, retrieved 2006-06-08 
5. ^ Shiga, David (2006-08-17), Mass cut-off between stars and brown dwarfs revealed, New Scientist, http://www.newscientist.com/article/dn9771-mass-cutoff-between-stars-and-brown-dwarfs-revealed.html, retrieved 2006-08-23 
6. ^ Basri, Gibor (2000), "Observations of Brown Dwarfs", Annual Review of Astronomy and Astrophysics 38: 485, Bibcode 2000ARA&A..38..485B, doi:10.1146/annurev.astro.38.1.485 
7. ^ Standish, E. M. (1998), JPL Planetary and Lunar Ephemerides, DE405/LE405, JPL IOM 312.F-98-048, http://iau-comm4.jpl.nasa.gov/de405iom/de405iom.pdf .
8. ^ ^{a b c} Pitjeva, E. V. (2005), "High-Precision Ephemerides of Planets—EPM and Determination of Some Astronomical Constants", Solar Syst. Res. 39 (3): 176–86, Bibcode 2005SoSyR..39..176P, doi:10.1007/s11208-005-0033-2, http://iau-comm4.jpl.nasa.gov/EPM2004.pdf .
9. ^ Vitagliano, A.; Stoss, R. M. (2006), "New mass determination of (15) Eunomia based on a very close encounter with (50278) 2000CZ12", Astron. Astrophys. 455 (3): L29–31, Bibcode 2006A&A...455L..29V, doi:10.1051/0004-6361:20065760 .
10. ^ Brown, Michael E.; Schaller, Emily L. (2007), "The mass of dwarf planet Eris", Science 316 (5831): 1585, Bibcode 2007Sci...316.1585B, doi:10.1126/science.1139415, PMID 17569855 .
11. ^ Tholen, David J.; Buie, Marc W.; Grundy, William M.; Elliott, Garrett T. (2008), "Masses of Nix and Hydra", Astron. J. 135 (3): 777–84, arXiv:0712.1261, Bibcode 2008AJ....135..777T, doi:10.1088/0004-6256/135/3/777 .
12. ^ ^{a b} Pitjeva, E. V.; Standish, E. M. (2009), "Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit", Celest. Mech. Dynam. Astron. 103 (4): 365–72, Bibcode 2009CeMDA.103..365P, doi:10.1007/s10569-009-9203-8, http://www.springerlink.com/content/21885q7262104u76/ .
13. ^ Ragozzine, Darin; Brown, Michael E. (2009), "Orbits and Masses of the Satellites of the Dwarf Planet Haumea = 2003 EL61", Astron. J. 137 (6): 4766–76, arXiv:0903.4213, Bibcode 2009AJ....137.4766R, doi:10.1088/0004-6256/137/6/4766 .
14. ^ IAU WG on NSFA Current Best Estimates, http://maia.usno.navy.mil/NSFA/CBE.html, retrieved 2009-09-25 .^{[dead link]}
15. ^ "The Final Session of the General Assembly", Estrella d'Alva: 1, 2009-08-14, http://www.astronomy2009.com.br/10.pdf .
16. ^ Anderson, John D.; Colombo, Giuseppe; Esposito, Pasquale B.; Lau, Eunice L.; Trager, Gayle B. (1987), "The Mass Gravity Field and Ephemeris of Mercury", Icarus 71 (3): 337–49, Bibcode 1987Icar...71..337A, doi:10.1016/0019-1035(87)90033-9 .
17. ^ Konopliv, A. S.; Banerdt, W. B.; Sjogren, W. L. (1999), "Venus Gravity: 180th Degree and Order Model", Icarus 139 (1): 3–18, Bibcode 1999Icar..139....3K, doi:10.1006/icar.1999.6086 .
18. ^ Konopliv, Alex S.; Yoder, Charles F.; Standish, E. Myles; Yuan, Dah-Ning; Sjogren, William L. (2006), "A global solution for the Mars static and seasonal gravity, Mars orientation, Phobos and Deimos masses, and Mars ephemeris", Icarus 182 (1): 23–50, Bibcode 2006Icar..182...23K, doi:10.1016/j.icarus.2005.12.025 .
19. ^ Jacobson, R. A.; Haw, R. J.; McElrath, T. P.; Antreasian, P. G. (2000), "A Comprehensive Orbit Reconstruction for the Galileo Prime Mission in the J2000 System", J. Astronaut. Sci. 48 (4): 495–516 .
20. ^ Stauch, R. A.; Antreasian, P. G.; Bordi, J. J.; Criddle, K. E.; Ionasescu, R.; Jones, J. B.; Mackenzie, R. A.; Pelletier, F. J. et al. (2006), "The gravity field of the Saturnian system from satellite observations and spacecraft tracking data", Astron. J. 132 (6): 2520–26, Bibcode 2006AJ....132.2520J, doi:10.1086/508812 .
21. ^ Jacobson, R. A.; Campbell, J. K.; Taylor, A. H.; Synott, S. P. (1992), "The Masses of Uranus and its Major Satellites from Voyager Tracking Data and Earth-based Uranian Satellite Data", Astron. J. 103 (6): 2068–78, Bibcode 1992AJ....103.2068J, doi:10.1086/116211 .
22. ^ Jacobson, R. A. (2009 April 3). "The Orbits of the Neptunian Satellites and the Orientation of the Pole of Neptune". The Astronomical Journal 137 (5): 4322–4329. Bibcode 2009AJ....137.4322J. doi:10.1088/0004-6256/137/5/4322.  edit
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See also

• Astronomical system of units
• Standard gravitational parameter